CN114491400A - Method for solving time-varying Siemens equation by noise suppression adaptive coefficient zero-ization neural network based on error norm - Google Patents
Method for solving time-varying Siemens equation by noise suppression adaptive coefficient zero-ization neural network based on error norm Download PDFInfo
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Abstract
The invention discloses a method for solving a time-varying Siemens equation by a noise suppression adaptive coefficient nulling neural network based on an error norm, which comprises the following steps of: a, firstly, establishing a mathematical model for solving a time-varying Siemens equation; b, defining a noise suppression adaptive coefficient zero neural network based on the error norm, and discussing the convergence of the noise suppression adaptive coefficient zero neural network; step C, adding different noises into the model, and discussing the robustness of the model under the influence of different noises; and D, initializing parameters, verifying and analyzing results, introducing a noise suppression self-adaptive coefficient based on an error norm into an ZNN model, theoretically analyzing the global convergence of the method, introducing different noises into the model, analyzing the stability of the model under the influence of different noises, and verifying that the model quickly converges to zero under the influence of different noises, thereby proving the effectiveness and superiority of the method.
Description
Technical Field
The invention relates to the technical field of time-varying Sieve's equation and neural network, in particular to a method for solving time-varying Sieve's equation based on an Adaptive coefficient nulling neural network (ACZNN for short) of noise suppression Adaptive coefficient based on error.
Background
The time-varying zerewith equation is an important branch of matrix theory. It has common applications in scientific research and engineering applications, such as robot kinematics, control theory, digital image processing, and communication engineering. Due to the important application of the time-varying siemens equation in many fields, it becomes more important to accurately solve the problem of the time-varying siemens equation.
In recent research and investigation, some numerical algorithms have been proposed to solve the above-mentioned problems. For example, it has been shown that newton's iteration method is used to calculate the time-varying sierster equation. Newton's iterative method is a classical numerical algorithm that solves the discrete time problem. From the control theory, newton's iteration method is a proportional feedback controller. Clearly, according to control theory, a controller using only proportional feedback cannot control a system with time-varying parameters in a predictive manner, resulting in hysteresis errors. In solving the zeroing problem, the Recurrent Neural Network (RNN) is typically designed as an ordinary differential function. The RNN model needs to evolve from an arbitrary initial value, continue in a given direction, and recursively compute an estimate for each point until it converges to the required accuracy. Therefore, the evolutionary direction of the algorithm needs to be modified according to the input state, forcing the residual to decrease to zero over time. The zero-ization neural network (ZNN) is an important component of the neural network as a parallel computing method and plays an important role in a linear computing problem or an optimization problem. For example, Liao et al propose an adaptive coefficient GNN model to solve the time-varying sierwise equation; gold et al propose a finite time recurrent neural network to solve this problem.
However, most of the neural networks are used for solving the time-varying sierwite equation at the present stage, but the convergence accuracy and the noise resistance of the neural networks have certain defects.
Disclosure of Invention
Based on the above discussion, the present invention proposes an error-based adaptive coefficient ZNN model, introduces the definition of error-based adaptive coefficients, and converts the coefficients of the OZNN model into error-related functions. The error-based adaptive coefficient ZNN model provided herein corrects the defect that the OZNN model cannot be stable under the influence of noise, that is, the corrected ZNN model still accurately solves the time-varying Sieve's equation for convergence within a finite time under the influence of noise, and the parallel computing model converts the problem of solving the time-varying Sieve's equation into a zero-solving problem of a linear equation for solving the time-varying Sieve's equation.
In order to achieve the purpose, the technical scheme adopted by the invention is as follows:
the method for solving the time-varying Siemens equation by the noise suppression adaptive coefficient zero-ization neural network based on the error norm comprises the following steps: the method comprises the following steps:
A. firstly, establishing a mathematical model for solving a time-varying Siemens equation;
B. defining an error norm-based adaptive nulling neural network, and discussing convergence;
C. adding different noises into the model, and discussing the robustness of the model under the influence of different noises;
D. initializing parameters, and verifying and analyzing results.
Preferably, the establishing of the mathematical model for solving the time-varying siemens equation in the step a includes the following steps:
A1. the Serveste equation is expressed as
A(t)X(t)-X(t)B(t)+C(t)=0
A2. Vectorizing two sides of the equation simultaneously to obtain
vec(A(t)X(t)-X(t)B(t))=-vec(C(t))
A3. From the kronecker product property:
wherein, the symbolRepresenting the kronecker product, thenx (t) vec (x (t)), b (t) vec (c (t)), and the following formula
P(t)x(t)+b(t)=0
Then the error function is written as
e(t)=P(t)x(t)+b(t)
A4. Obtained by OZNN model
Preferably, the specific process of defining the error norm-based adaptive nulling neural network and determining its convergence in step B is as follows:
B1. the adaptive coefficient based on the error norm is specifically
a. Exponential adaptive coefficient
b. Logarithmic adaptive coefficient
λ(e(t))=r|log2||e(t)||2|+r,
c. Fractional adaptive coefficient
Wherein r >1 is a constant;
B2. defining an error norm based noise suppression adaptive coefficient zero-ization neural network as follows:
where μ >0 is a constant and phi (-) is expressed as an excitation function, expressed specifically as
B3. Order to
Make it derived from time
Thus, can obtain
Defining a Lyapunov candidate function as
Wherein κ>0, obtaining Gi(t) is positive, GiThe time derivative of (t) is described as
The following two formulas
Can be obtained by finishing
According to the median theorem in the mathematical theory, the method obtains
Since the excitation function is a monotonically increasing function, it is possible to use a single excitation functionSpecifying a section When q isi(t)∈D1When it is used, makeThen the following inequality is:
|φ(qi(t))|≤S1|qi(t)|
|∈i(t)φ(qi(t))|≤S1|∈i(t)||qi(t)|
Is similarly obtained
|φ(qi(t))|≥S3|qi(t)|
|φ(∈i(t))|≤S2|∈i(t)|
In view of the above, it is desirable to provide,
Preferably, the step C of determining the model of the introduced noise and the stability thereof includes the following specific steps:
C1. after the introduction of noise, the basic model is
C2. Xi is ai(t) is constant noise, then obtain
The Lyapunov candidate function is defined as follows:
derivation of this can yield:
from the above analysis, it can be known that the model still maintains stability under the influence of constant noise;
C3. xi is ai(t) is linear noise, then there is the following derivation: defining a function
The following cases are discussed:
b.ξi(t)qiwhen (t) ≧ 0, when | qi(t) when increasing, | - μ φ (q) |i(t))+ξi(t) | will decrease until- μ φ (q)i(t))+ξi(t)=0;
At this time ui(t) obtaining a minimum value, thereby obtaining the following inequality
Because of | phi-1(.) | is less than or equal to |, so
In case of a certain moment t1Satisfy the equationAt the same time have t2Meet at the momentAnd t is2-t1If at t, δ1To t2The moment always existsThen there is
In combination with the above two formulas,
The left side of the above equation is an increasing function with respect to δ, and the right side is a fixed value, resulting in
When in useThen, similarly, obtain Finally, the error norm converged toThe model therefore proves to be stable also in the presence of disturbances of the linear noise.
Preferably, the result verification and analysis in step D specifically comprises the following steps:
D1. given as an example
And adjusting the values of r and mu, substituting the values into the model, so that the experimental result accords with the expectation, and proving the effectiveness of the model.
The invention has the beneficial effects that: the invention discloses a method for solving a time-varying Siemens equation by a noise suppression adaptive coefficient zero-ization neural network based on an error norm, which has the improvement that:
1) an error-based adaptive coefficient is defined, the limit that an original ZNN model coefficient is a constant is broken through, and a variable coefficient related to the error is changed, so that the error-based adaptive coefficient is better adapted to the variable condition in real life.
2) The defect that the original ZNN model cannot be stable under noise interference is corrected by the error norm-based noise suppression adaptive coefficient zero-ization neural network, namely the corrected ZNN model still accurately solves the time-varying Siemens equation under the condition of noise interference.
3) The parallel computation model converts the time-varying Sieve's equation into a zero-solving problem of a linear equation for solving the time-varying Sieve's equation.
Drawings
FIG. 1 is a flow chart of the ACZNN method of the present invention.
FIG. 2 is a diagram illustrating a comparison between a computed solution and a theoretical solution trajectory using the ACZNN model, according to an embodiment of the present invention. (ii) a
Fig. 3 shows simulation results of ACZNN model using three different adaptive coefficients when r ═ μ ═ 3 according to an embodiment of the present invention.
Fig. 4 shows simulation results of the ACZNN model under the influence of constant noise when r ═ μ ═ 3 in the embodiment of the present invention.
Fig. 5 shows simulation results of the ACZNN model under the influence of linear noise when r ═ μ ═ 3 according to an embodiment of the present invention.
Detailed Description
In order to make those skilled in the art better understand the technical solution of the present invention, the following further describes the technical solution of the present invention with reference to the drawings and the embodiments.
A method for solving a time-varying sierster equation for a noise suppression adaptive coefficient nulling neural network based on an error norm as described with reference to fig. 1-5, comprising:
A. firstly, establishing a mathematical model for solving a time-varying Siemens equation;
A1. the Serveste equation is expressed as
A(t)X(t)-X(t)B(t)+C(t)=0
A2. Vectorizing two sides of the equation simultaneously to obtain
vec(A(t)X(t)-X(t)B(t))=-vec(C(t))
A3. From the kronecker product property:
wherein, the symbolRepresenting the kronecker product, thenx (t) vec (x (t)), b (t) vec (c (t)), and the following formula
P(t)x(t)+b(t)=0
Then the error function is written as
e(t)=P(t)x(t)+b(t)
A4. Obtained by OZNN model
B. Defining an error norm-based adaptive nulling neural network, and discussing convergence;
B1. the adaptive coefficient based on the error norm is specifically
a. Exponential adaptive coefficient
b. Logarithmic adaptive coefficient
λ(e(t))=r|log2||e(t)||2|+r,
c. Fractional adaptive coefficient
Where r >1, is a constant.
B2. Defining an error norm based noise suppression adaptive coefficient zero-ization neural network as follows:
where μ >0, is a constant. Phi (-) is expressed as an excitation function, expressed specifically as
B3. Order to
Make it derived from time
Thus, can obtain
Defining a Lyapunov candidate function as
Wherein κ>0, obtaining Gi(t) is positive. GiThe time derivative of (t) is described as
The following two formulas
Can be obtained by finishing
According to the median theorem in the mathematical theory, the method obtains
Since the excitation function is a monotonically increasing function, it is possible to use a single excitation functionSpecifying a section D1=When q isi(t)∈D1When it is used, orderThen the following inequality is:
|φ(qi(t))|≤S1|qi(t)|
|∈i(t)φ(qi(t))|≤S1|∈i(t)||qi(t)|
Is similarly obtained
|φ(qi(t))|≥S3|qi(t)|
|φ(∈i(t))|≤S2|∈i(t)|
In view of the above, it is desirable to provide,
C. Adding different noises into the model, and discussing the robustness of the model under the influence of different noises;
C1. after the introduction of noise, the basic model is
C2. Xi is ai(t) is constant noise, then obtain
The Lyapunov candidate function is defined as follows:
derivation of this can yield:
from the above analysis, it can be seen that the model still maintains stability under the influence of constant noise.
C3. Xi is ai(t) is linear noise, then there is the following derivation: defining a function
The following cases are discussed:
b.ξi(t)qiwhen (t) ≧ 0, when | qi(t) when increasing, | - μ φ (q) |i(t))+ξi(t) | will decrease until- μ φ (q)i(t))+ξi(t)=0;
At this time ui(t) taking the minimum value. The following inequality is obtained
Because of | phi-1(. h) | is less than or equal to |, so
In case of a certain moment t1Satisfy the equationAt the same time have t2Meet at the momentAnd t is2-t1If at t, δ1To t2The moment always existsThen there is
In combination with the above two formulas,
The above equation is an increasing function with respect to δ to the left and a fixed value to the right. To obtain
When in useThen, similarly, obtain Finally, the error norm converged toThe model therefore proves to be stable also in the presence of disturbances of the linear noise.
D. Initializing parameters, and verifying and analyzing results.
D1. Given as an example
And adjusting the values of r and mu, substituting the values into the model, so that the experimental result accords with the expectation, and proving the effectiveness of the model.
Examples
The method for solving the time-varying Siemens equation based on the zero-ization neural network is utilized for calculation: examples are as follows:
(1) in this example, r ═ μ ═ 3.
(2) And (4) respectively substituting the examples into ACZNN models with different adaptive coefficients for calculation. The comparison and error map of the calculated solution and the theoretical solution trajectory are shown in fig. 2 and fig. 3 (fig. 2 is a comparison map between the calculated solution and the theoretical solution trajectory of the ACZNN model, fig. 3(a) is a residual map under the influence of exponential adaptive coefficients, fig. 3(b) is a residual map under the influence of logarithmic adaptive coefficients, and fig. 3(c) is a residual map under the influence of fractional adaptive coefficients). Under the influence of noise, the error map of the ACZNN model for solving the target problem is shown in fig. 4 and fig. 5 (fig. 4(a) is a linear residual map under the action of constant noise, fig. 4(b) is a d-log residual map under the action of constant noise, fig. 5(a) is a linear residual map under the action of linear noise, and fig. 5(b) is a log residual map under the action of linear noise), and it is seen from the diagram that under the action of the ACZNN model, the calculated solution quickly converges to the theoretical solution, and the error also quickly converges to zero. Thereby proving the effectiveness and superiority of the ACZNN model.
The principal features of the invention and advantages of the invention have been shown and described. It will be understood by those skilled in the art that the present invention is not limited to the embodiments described above, and that various changes and modifications may be made without departing from the spirit and scope of the invention as defined in the appended claims. The scope of the invention is defined by the appended claims and equivalents thereof.
Claims (5)
1. The method for solving the time-varying Siemens equation by the noise suppression adaptive coefficient zero-ization neural network based on the error norm comprises the following steps: the method is characterized by comprising the following steps:
a, firstly, establishing a mathematical model for solving a time-varying Siemens equation;
b, defining a noise suppression adaptive coefficient zero neural network based on the error norm, and judging the convergence of the noise suppression adaptive coefficient zero neural network;
step C, adding different noises into the model, and discussing the robustness of the model under the influence of different noises;
and D, initializing parameters, and verifying and analyzing results.
2. The method for solving the time-varying siemens equation based on the error norm noise suppression adaptive coefficient nulling neural network of claim 1, wherein said establishing a mathematical model for solving the time-varying siemens equation of step a comprises the steps of:
A1. the Serveste equation is expressed as
A(t)X(t)-X(t)B(t)+C(t)=0
A2. Vectorizing two sides of the equation simultaneously to obtain
vec(A(t)X(t)-X(t)B(t))=-vec(C(t))
A3. From the kronecker product property:
P(t)x(t)+b(t)=0
Then the error function is written as
e(t)=P(t)x(t)+b(t)
A4. Obtained by OZNN model
3. The method according to claim 1, wherein the step B of defining the error norm based noise suppression adaptive coefficient nulling neural network comprises:
B1. the adaptive coefficient based on the error norm is specifically
a. Exponential adaptive coefficient
b. Logarithmic adaptive coefficient
λ(e(t))=r|log2||e(t)||2|+r,
c. Fractional adaptive coefficient
Wherein r >1 is a constant;
B2. defining an error norm based noise suppression adaptive coefficient zero-ization neural network as follows:
where μ >0 is a constant and phi (-) is expressed as an excitation function, expressed specifically as
B3. Order to
Make it derived from time
Thus, can obtain
Defining a Lyapunov candidate function as
Where κ >0, to give Gi(t) is positive, GiThe time derivative of (t) is described as
Can be obtained by finishing
According to the median theorem in the mathematical theory, the method obtains
Since the excitation function is a monotonically increasing function, it is possible to use a single excitation functionSpecifying a section When q isi(t)∈D1When it is used, orderThen the following inequality is:
|φ(qi(t))|≤S1|qi(t)|
|∈i(t)φ(qi(t))|≤S1|∈i(t)||qi(t)|
Is similarly obtained
|φ(qi(t))|≥S3|qi(t)|
|φ(∈i(t))|≤S2|∈i(t)|
In view of the above, it is desirable to provide,
4. The method for solving the time-varying siemens equation based on the error norm noise suppression adaptive coefficient nulling neural network of claim 1, wherein the noise addition model of step C is determined by the specific steps of:
C1. after the introduction of noise, the basic model is
C2. Xi is ai(t) is constant noise, then obtain
The Lyapunov candidate function is defined as follows:
derivation of this can yield:
from the above analysis, it can be seen that under the influence of constant noise, the model still maintains stability,
C3. xi is ai(t) is linear noise, then there is the following derivation: defining a function
The following cases are discussed:
b.ξi(t)qiwhen (t) ≧ 0, when | qi(t) when increasing, | - μ φ (q) |i(t))+ξi(t) | will decrease until- μ φ (q)i(t))+ξi(t)=0;
At this time ui(t) obtaining a minimum value, thereby obtaining the following inequality
Because of | phi-1(.) | is less than or equal to |, so
In case of a certain moment t1Satisfy the equationAt the same time have t2Meet at the momentAnd t is2-t1If at t, δ1To t2The moment always existsThen there is After that have
In combination with the above two formulas,
The left side of the above equation is an increasing function with respect to δ, and the right side is a fixed value, resulting in
5. The method for solving the time-varying siemens equation based on the noise suppression adaptive coefficient nulling neural network of the error norm as set forth in claim 1, wherein the result verification and analysis in step D specifically comprises the steps of:
D1. given as an example
And adjusting the values of r and mu, substituting the values into the model, so that the experimental result accords with the expectation, and proving the effectiveness of the model.
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